A multilinear Nyström algorithm for low-rank approximation of tensors in Tucker format

The Nystr"om method offers an effective way to obtain low-rank approximation of SPD matrices, and has been recently extended and analyzed to nonsymmetric matrices (leading to the generalized Nystr"om method). It is a randomized, single-pass, streamable, cost-effective, and accurate alternative to the randomized SVD, and it facilitates the computation of several matrix low-rank factorizations. In this paper, we take these advancements a step further by introducing a higher-order variant of Nystr"om's methodology tailored to approximating low-rank tensors in the Tucker format: the multilinear Nystr"om technique. We show that, by introducing appropriate small modifications in the formulation of the higher-order method, strong stability properties can be obtained. This algorithm retains the key attributes of the generalized Nystr"om method, positioning it as a viable substitute for the randomized higher-order SVD algorithm.